Leveraged ETFs deserve to be judged by realized wealth and expected log growth, and Murray and Sammon make the cleanest case I know. Almost every demonstration survives scrutiny. Choosing how much leverage to hold remains outside the paper's remit, as the authors acknowledge. The abstract presents a measurement paper. Even with financing at the risk-free rate, their moderate-leverage simulations trail the unlevered market from 2000 to 2009.
Start with the wealth identity
For a costless, continuously rebalanced 2x position, the paper starts with an exact identity: terminal wealth per dollar equals (1 + 2R + R²) times exp(−RV). R denotes the underlying's cumulative simple return. RV is realized variance, calculated as the sum of squared log returns along the path. The authors name R² the convexity bonus. Always nonnegative, it increases with the size of a move in either direction. Volatility drag enters through exp(−RV), multiplying the entire payoff, including the bonus. At general leverage L, its coefficient is L(L−1)/2. The coefficient is therefore 1 at 2x and 3 at 3x, with the same values applying to −1x and −2x inverse funds.
Consider consecutive days returning 10% each. Doubling the simple return produces 42%; the costless 2x fund earns 44%, with a 4.41 point bonus offset by 2.41 points of drag. NVIDIA supplies a larger example in 2024. Its stock gained 171%, while NVDX gained 382.25%. That exceeds the 342% obtained by simple doubling and falls short of the costless estimate of 455.2%. The calculation contains a 292.4 point bonus and a 0.756 drag multiplier.
The authors then put the familiar performance measures through their tests. Their main target, the compounded sample-mean benchmark, compounds a fund's historical arithmetic mean return over the holding period. XIV makes the problem plain. From November 2011 to February 15, 2018, its arithmetic mean daily return was 0.234%. Compound that mean and $1 becomes $40.25. Compound the returns investors actually received and it becomes $0.94.
Expected terminal wealth fails differently. Under geometric Brownian motion (GBM), the authors use daily log volatility of 5%, roughly 79% annualized, an extreme calibration. A $10,000 2x position then produces mean 20-year wealth of $167.18 billion. Median wealth is $1.90.
Resampling shifts 3x expected wealth by a factor of 15.28. The authors draw from a century of CRSP daily returns, using individual days or whole blocks. In the July to June panel, a $10,000 position at 3x has expected 20-year wealth of $27,050,302 under annual blocks. With two-decade blocks, it has $1,770,639. Every scheme leaves the log-based measure, exp E[log W], at $466,770.
The Korean case studies follow investor dollars rather than fund returns. Across ten LETFs, cumulative investor P&L reached $34.8 billion on June 22, 2026, then fell to −$193 million by September 1. Replaying those same flows along a smooth geometric-mean path gives a timing-neutral benchmark profit of $4.711 billion. Closing-auction hedging contributes directly to drag as well. The paper calculates that a one-dollar impact bump costs a 2x fund 0.1654 points even when the bump never reverses.
Which return should count?
The argument fits inside the two-period toy example. The underlying travels from 100 to 101 to 100, leaving the costless 2x fund at $99.9802. Its arithmetic mean daily return is 0.0099%; compounding that mean produces $100.0198. Three numbers describe the same path. Realized wealth declines, the arithmetic benchmark rises, and average log return is negative.
The authors argue that a preference for log growth requires no log-utility assumption. Stationary, ergodic returns with positive gross returns suffice: the time average of log returns converges along a realized path. I accept that account of long-run compounding. Choosing the criterion still leaves prediction untouched. E[log W] requires estimates of μ and σ.
Their bias table puts a price on using arithmetic means. The simulation uses iid lognormal daily returns, with a 40-year calibration and a 20-year forecast. Bias in the log forecast is zero. For every block length shorter than the 20-year horizon, the arithmetic benchmark's bias ranges from 210.9% to 238.1%. It reaches zero only when the block matches that horizon.
Unbiased remains a long way from accurate.
Where L² overtakes Lμ
The paper works with a leading-order growth rate: E[log(1 + Lr)] ≈ Lμ − (L²/2)E[r²]. Its simulations add financing on the borrowed (L−1) and a fee. Exposure earns Lμ, against costs of (L−1)r_f, the fee, and a variance term growing with L². The convexity bonus belongs inside realized (G_T)^L. It pays on paths with large cumulative moves.
Beating the underlying and selecting optimal leverage are different tasks. The breakeven return addresses the former: it is the cumulative move at which the LETF matches the unlevered asset. That threshold rises with costs and the sum of squared returns. As for optimal leverage, the paper says expected log wealth, like realized compound wealth, can peak well below the wipeout threshold. It specifies no ratio.
With financing alone, Table 11 reports CAGR of 11.95% at 1x, 16.70% at 2x and 17.94% at 3x from August 1976 to 2024. Under the 1.47% fee sensitivity, the 3x position also suffers a −97.9% maximum drawdown.
The funding assumption does much of the work
Table 12 offers the practical case for moderate leverage. The authors simulate daily-reset 1.25x and 1.5x exposure to the Fama-French market, charging a 0.05% fee and financing at the risk-free rate. From August 1976 to December 2024, CAGR comes to 11.95%, 13.39% and 14.68% for 1x, 1.25x and 1.5x. The corresponding maximum drawdowns are −54.7%, −64.2% and −72.0%. Over February 2010 to December 2024, those columns return 14.34%, 17.27% and 20.08%. For 2000 to 2009, they return −0.38%, −1.97% and −3.78%.
The authors describe risk-free financing as "deliberately favorable" and say the paths "do not identify an optimal leverage ratio." Their affirmative claim goes only as far as showing the conditions under which moderate leverage looks more plausible as a long-run product. Their figures leave little room for extra costs. In Table 11's 1.47% fee sensitivity, the return at 1.25x falls to 11.79%. The paper acknowledges that this puts it slightly behind the market's 11.95%. Funding spreads eat into the same margin, and Table 12 leaves them out.
The 2x mutual funds illustrate the cumulative cost. Expenses reduce ULPIX terminal wealth by 32.8%, with risk-free funding taking another 37.1%. Adding back both still leaves ULPIX ending 72.7% below the squared-underlying benchmark. The paper attributes that combined gap to volatility drag, funding spreads, trading costs and tracking differences.
We tried a momentum selector on SPY, QQQ and their 2x and 3x funds
The paper tests no timing strategy. We derived the rolling allocation rule below as an application; the authors establish nothing about whether it makes money. Their historical fund-return examples compare performance. We did not rerun Table 12, which requires fixed-leverage daily resets on the Fama-French market with risk-free financing. Our test uses a different approach and market universe.
At each month-end, the rule ranks SPY, SSO and UPRO by mean daily log return over the trailing 252 sessions. It separately ranks QQQ, QLD and TQQQ on the same basis. At the next session's open, it allocates half of wealth to each family's winner. The 252-day lookback and monthly schedule are our choices. Scoring actual fund prices incorporates embedded fees and financing into the score. We also charged $0.004 a share, subject to a $1 minimum. Our metrics cover January 4, 2021 to July 1, 2024.
This tests trailing momentum as a way to choose leverage, a claim the paper never makes. Its nearest comparison is 13.39% CAGR at fixed 1.25x and 14.68% at 1.5x over 1976 to 2024. Method, universe and period all differ, so a gap between the results tells us nothing about replication.
Our selector earned a total return of 89.65% from January 4, 2021 to July 1, 2024. Sharpe was 0.65, Sortino 0.89 and Calmar 0.28. Volatility reached 39.79%; maximum drawdown was -72.27%. That loss almost matches the paper's -72.0% drawdown for fixed 1.5x over 1976 to 2024. Their path needed 48 years and two bear markets to reach it. Ours needed 42 months. The paper's 13.39% and 14.68% are CAGRs for a different window, measuring something different from our total return.
Read the 39.79% volatility before the 89.65% return. With permission to hold 3x funds in both halves, a selector will owe most of its result to exposure. Forty-two months covers one regime. This was one quick automated test of our own rule. Whether its result looks weak or strong says more about that rule than about the paper.
What would turn the measurement into a leverage rule?
Choosing leverage comes after the work this paper does. Its leverage backtests keep L fixed throughout each window. A strategy would need trailing estimates of μ, σ and a funding spread, along with a leverage cap. The bias table shows how far a trailing estimate can wander.
For 3x in the July-June panel, expected wealth differs by 15.28x between annual and two-decade blocks. Move the year boundary from January to July, also shifting the sample window by six months, and 3x annual-block wealth changes from $10,027,175 to $27,050,302. E[log W] remains unchanged. That invariance applies only to the mean of log wealth. Drawdown risk depends on the dependence structure retained by the blocks, and the authors disclose that their two-decade model uses five historical blocks.
I would change my view for a rule that chooses leverage from trailing estimates and beats 1x through 2000 to 2009 after a realistic spread.
Our backtest stops at 2024-07-01, and everything after that date is deliberately left untouched so the same strategy can be checked out of sample later.